Problem 88 - College Algebra

Question

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ h(x)--|x+3| $$

Step-by-Step Solution

Answer

The function \( h(x) = |x + 3| \) is a horizontal shift of the base function \( f(x) = |x| \), moved 3 units to the left. This translates every point on the original graph 3 units towards the left, with the vertex of the V shape moving from (0,0) to (-3,0).

Step 1: Understand the base function

Begin by understanding the base function \( f(x) = |x| \). The graph of this function is a V shape that intersects the origin of the coordinate plane. The point of the V is at the origin, (0,0), and it opens upward with slopes of 1 and -1.

Step 2: Identify the transformation

The function \( h(x) = |x + 3| \) represents a horizontal translation (or shift) of the base absolute function. The +3 inside the absolute value shifts the graph 3 units to the left. Hence, every point on \( f(x) = |x| \) moves 3 units towards the left to form \( h(x) = |x + 3| \).

Step 3: Graph the transformed function

Take each point on \( f(x) = |x| \) and shift it 3 units to the left. The vertex moves from (0,0) to (-3,0). The graph will still retain its V shape, but the vertex will now be placed at (-3,0).

Key Concepts

Graph Transformations

Graph transformations are processes that alter the appearance of a graph of a function without affecting its fundamental shape. These transformations allow us to modify the position, orientation, and size of the graph on the coordinate plane. In general, transformations can be categorized into several types: translations, reflections, stretches, and shrinkages. Each type alters the graph in a different but predictable way.

• **Translations** move the graph horizontally or vertically without changing its shape.
• **Reflections** flip the graph over a specific axis.
• **Stretches** and **shrinkages** change the graph's size either by pulling it away or pushing it towards an axis.

Graph transformations are key tools because they enable us to understand various forms of functions by adjusting a base graph, like the absolute value function, to match another function's graph. By using transformations, complex or shifted graphs become more manageable and can be analyzed more easily.

Horizontal Translation

Horizontal translation involves shifting the graph of a function left or right along the x-axis. This shift is determined by the constant added or subtracted inside the function. When dealing with an absolute value function like \( h(x) = |x + 3| \), the expression inside the absolute value sign indicates the direction and magnitude of the shift.

- **Right Shift**: The graph moves to the right if a negative constant is added (e.g., \( x - c \)).- **Left Shift**: The graph shifts to the left if a positive constant is added (e.g., \( x + c \)).
In this context, \( |x + 3| \) means the graph of the base function \( |x| \) is translated 3 units to the left. This shift doesn't alter the V shape of the absolute value function; it only moves the entire graph horizontally, including all the points on it.

Vertex of a Graph

The vertex of a graph, especially for the absolute value function, is a crucial point that signifies the peak or the lowest point of the graph. For the absolute value function \( f(x) = |x| \), the vertex is originally at the origin, at point \((0, 0)\). The vertex is where the graph changes direction, forming a sharp point, particularly in the V shape of an absolute value graph.

In the function \( h(x) = |x + 3| \), after the horizontal translation is applied, the vertex shifts accordingly. The vertex moves to \((-3, 0)\) due to the 3-unit shift to the left. While other points on the graph are also shifted, the vertex is particularly significant because it is the central point around which the symmetry of the absolute value graph remains centered. Understanding the location of the vertex helps in sketching accurate graphs and determining the turning points and symmetry in functions.